Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11717309 |
| commutator, cosocle | C2×#1039 | C23 |
| Frattini, hypercenter | C22 | (D5⋊AGL(1,𝔽5))⋊C2 |
| Fitting | C22×C52 | #41 |
| socle | C2×C52 | #121 |
| hypocenter | C52 | #472 |
| LCS | G ▷ C2×#1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
185 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^6,5^2|b^2=d,d^2=f,c^a=cde,e^a=ef,g^a=g^4,c^b=cd,g^b=g^2,h^b=h^3,d^c=df,g^c=h^4,h^c=g^4,g^d=g^4,h^d=h^437 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5744);